Converting a 9×13″ pan recipe to a 10″ round pan
Batter scales with volume, not with width. A 9×13″ pan holds about 14 cups filled to the brim, and a 10″ round pan holds 11 cups, so you need 21% less batter. Multiply every ingredient — including the eggs, which is the awkward one — by 0.79×.
The number people miss is depth. Two pans can hold the same volume and still bake completely differently, because what matters is how thick the batter sits: volume divided by bake surface. Here that moves from about 1.2″ to 1.3″. Batter depth is similar in both pans. Bake time should be close to the original; start testing 5 minutes early.
How many pans one batch fills
A 9×13″ pan recipe comes to a little over one 10″ round — about 1¼ pans' worth — so a single pan would be overfilled. Scale the recipe by 0.79× to fill one pan to two-thirds, or bake the extra batter separately as a few cupcakes.
Cross-check against Wilton's chart: Wilton puts 10 cups of batter in one 2-inch layer of a 9×13″ pan and 6 in a 10″ round pan, a ratio of 1⅔ — against 1¼ by brim capacity. The two differ because Wilton fills each pan to its own level (anywhere from half to two-thirds); this converter uses brim capacity throughout so every pair is on the same footing.
A caution worth reading
Volume conversions hold well for cakes, brownies and quick breads — anything poured. They hold less well for recipes where structure comes from the pan itself: a delicate chiffon or angel food climbs the ungreased side of a tube pan, and a loaf's shape is part of how it bakes. For those, scale the ingredients by all means, but expect the texture to shift, and use the skewer rather than the recipe's clock.